the pattern so they are excluded. Also like before
the connects has 2 layers so we use this original
equation.
C=aH+b
We use the same method as before.
9=a3+b (1)
24=a4+b (2)
15=a (2)-(1)
Substitute ‘a’ which is 15 back into (1)
9=15x3+b
9=45+b
b= -36
Substitute ‘a’ and ‘b’ back into original equation
C=15H-36 that is the equation for the number
of connects in a Nx6 box. But since the
first 2 heights didn’t follow the
pattern we didn’t use them in the
equation so this equation doesn’t
work for them.
Connect 4
Hx4 Box
Hx4 1 2 3 4 5 6
Connects 1 2 3 10 17 24 first layer
7 7 7 second layer
Like before the first to equations do not follow the
pattern so they are excluded. Also like before the
connects has 2 layers so we use this original
equation.
C=aH+b
We use the same method as before.
3=a3+b (1)
10=a4+b (2)
7=a (2)-(1)
Substitute ‘a’ back into (1)
3=7x3+b
3=21+b
b= -18
Substitute ‘a’ and ‘b’ back into original equation
C=7H-18
that is the equation for the number of connects in a Nx4
box. But since the first 2 heights didn’t follow the
pattern we didn’t use them in the equation so this
equation doesn’t work for them.
Formula For Connect 4
The formula for any box with a width of 4 is C=7H-18
The formula for any box with a width of 5 is C=11H-27
The formula for any box with a width of 6 is C=15H-36
There is a visible pattern the first number always goes up by 4 and the second number follows the 9 times table.
F=First number
S=Second number
W Formula F S
4 7H-18 7 18
5 11H-27 11 27
6 15H-36 15 36
Again the same equation is used as there are 2 layers to the first number.
F=aW+b
Because there are 2 layers we put in the ‘First number’ twice.
7=a4+b (1)
11=a5+b (2)
4=a (2)-(1)
Substitute ‘a’ back into (1)
7=4x4+b
7=16+b
B= -9
Substitute ‘a’ and ‘b’ back into original equation
F=4W-9 that is the equation for getting the first number.
The second number has two layers so again the equation is the same as the first numbers equation.
S=aW+b
And because there are 2 layers we put in the second number twice.
18=a4+b (1)
27=a5+b (2)
9=a (2)-(1)
Substitute ‘a’ back into (1)
18=9x4+b
18=36+b
b= -18
Substitute ‘a’ and ‘b’ back into second number equation
S=9w-18
So we have the equations for the first and second numbers of the table. So now we put them together.
The first number equation is solved by 4W-9 and then to that you had to times the first number by H,
(4W-9)H
In the equations for nxW you had to times the first number by H then you had to minus the second number. This means you have to minus the second number equation.
(4W-9)H-(9W-18)
That is the equation for any number of connects in any size box when you are using connect 4.
To make the equation simpler, we can multiply out the brackets and this results in the equation 4Wh-9H-9W+18.
Connect 3
Hx6 Box
Hx6 1 2 3 4 5 6 first layer
Connects 4 8 26 44 62 80 second layer
There are the same number of layers as in connect
4 so the entire equation has the same format.
This time in connect 3 the only box that dose not
Fit the formula is the first box.
C=aH+b
8=a2+b (1)
26=a3+b (2)
18=a (2)-(1)
Substitute back into (1)
8=18x2+b
8=36+b
b=-28
Then we Substitute back into original equation
C=18H-28 is the formula for Nx6 box when you
are using connect 3. But since the height 1 box
doesn’t follow the pattern, then this equation doesn’t
work for that box.
Formula For Connect 3
The formula for any box with a width of 4 is C =10H-16
The formula for any box with a width of 5 is C=14H-22
The formula for any box with a width of 6 is C=18H-28
There is a visible pattern the first number always goes up by 4 and the second number goes up by 6.
W Formula F S
4 10H-16 10 16
5 14H-22 14 22
6 18H-28 18 28
Again I use the Substitution method that I used to find the connect 4 formula. This produced two parts of equations. I got the result F=4W-6 for the first number equation. And I got the result S=6W-8 for the second number. Put the 2 equations together and get (4W-6)H-(6W-8), this can be made simpler, it goes to 4HW-6H-6W+8. That is the formula for finding the number of connects in any size box when you are connecting 3.
Formula For Connect 5
As you can see I only followed what I did in connect 4 and 3 for the width of 5 and 4. I noticed in my both my connect 4 and connect 3 equations, that there was an increase in every formula. I will use connect 3 as an example
W Formula F S
4 10H-16 10 16
5 14H-22 14 22
6 18H-28 18 28
In the formula column, the first part of the equation increases by 4 each time. The second part is increased by 6 every time.
W Formula F S
4 7H-18 7 18
5 11H-27 11 27
6 15H-36 15 36
In the formula column, the first part of the equation increases by 4 each time. The second part is increased by 9 every time.
Now that I have the first 2 parts of connect 5 I can use them to get the next formula.
W Formula
4 C =4H-16
5 C =8H-28
The increase for the first number is 4. the increase for the second number is 12. Therefore I know that the next formula is C =12H-40
Again I use the Substitution method that I used to find the connect 5 formula. This produced two parts of equations. (4W-12)H-(12W-32), which I simplified to 4Wh-12H-12W+32. This is the formula for finding the number of connects in any size box when you are connecting 5.
I then realised that if the formula doesn’t work for a box which has the height which is 2 numbers lower than the number that is being connected. E.g if your connecting 4 then the formula doesn’t work for a box with a height of 2. It doesn’t matter what size the width is. This is because these heights don’t follow the pattern that the other heights over 2 in the case connect 4 do.
Any Size Box And Any Number Of Connects
I now have the formulas for any size box and when your connecting 3, 4 and 5. I put them in this table.
T=Third number
Fo=Fourth number
Connects Formula F S T Fo
3 (4W-6)H-(6W-8) 4 6 6 8
4 (4W-9)H-(9W-18) 4 9 9 18
5 (4W-12)H-(12W-32) 4 12 12 32
The first number is always 4.
To get the second part of the equation we need to find the increase. The second number always goes up by three so it has 2 layers so we use this equation.
S=aC+b
As there are 2 layers we put in two results.
6=a3+b (1)
9=a4+b (2)
3=a (2)-(1)
Substitute ‘a’ back into (1)
6=3x3+b
6=9+b
b=-3
Substitute ‘a’ and ‘b’ back into original equation.
S=3C-3
This means the second number equation is produced by S=3C-3
And since the third number has exactly the same increase as the second number the equation is exactly the same.
So the third number equation is T=3C-3
The fourth number is different
Fourth number
8 18 32
10 14
4
It goes up by 10 then 14. That is an increase of 4. So to the fourth number there are 3 layers.
So the equation we use is Fo=aC²+bC+z
Since the equation has three layers to it has three unknowns a, b and z so we have to put in 3 results.
8=a3²+b3+z (1)
18=a4²+b4+z (2)
32=a5²+b5+z (3)
8=a9+b3+z (1)
18=a16+b4+z (2)
32=a25+b5+z (3)
14=a9+b (3)-(2) (4)
10=a7+b (2)-(1) (5)
4=2a (4)-(5)
2=a
Substitute ‘a’ back into (4)
14=2x9+b
14=18+b
b=-4
Substitute ‘a’ and ‘b’ into (3)
32=2x25-4x5+z
32=50-20+z
32=30+z
z=2
So we now know what ‘a’, ‘b’ and ‘z’ are so we sub them back into the original equation which was Fo=aC²+bC+z
Fo=2C²-4C+2 is the fourth number equation
So the first number equation is 4
The second number equation is 3C-3
The third number equation is 3C-3
The fourth number equation is Fo=2C²-4C+2
So we now join them together.
In the connect number equations the first 2 numbers were in brackets and so were the second 2. So we have to group the first 2 equations and the second 2 in brackets. But each equation has to be in its own brackets so we need to use double brackets.
The formula for connect 3 was (4W-6)H-(6W-8) we will use it as a bass.
(first number equation xW-second number equation)H-(third number equation-Fourth number equation)
But each number equation needs to be surrounded by its own brackets.
((first number equation xW)-(second number equation))H-((third number equation)xW-(Fourth number equation))
((4W)-(3C-3))H-((3C-3)xW-(2C²-4C+2))
This formula finds out how many connects there are in any size box using any connecting number. E.g you could use a height of 5 and width of 4 and we are connecting 4. This would give you the answer of 17 which is correct.
But as before the formula does not work if the height is 2 or more numbers lower than the number you are connecting.